1math::combinatorics(n)         Tcl Math Library         math::combinatorics(n)
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NAME

8       math::combinatorics - Combinatorial functions in the Tcl Math Library
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SYNOPSIS

11       package require Tcl  8.2
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13       package require math  ?1.2.3?
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15       ::math::ln_Gamma z
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17       ::math::factorial x
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19       ::math::choose n k
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21       ::math::Beta z w
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DESCRIPTION

26       The  math  package contains implementations of several functions useful
27       in combinatorial problems.
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COMMANDS

30       ::math::ln_Gamma z
31              Returns the natural logarithm of  the  Gamma  function  for  the
32              argument z.
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34              The Gamma function is defined as the improper integral from zero
35              to positive infinity of
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37                t**(x-1)*exp(-t) dt
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39
40              The approximation used in the Tcl Math Library is from  Lanczos,
41              ISIAM  J. Numerical Analysis, series B, volume 1, p. 86.  For "x
42              > 1", the absolute error of the result is claimed to be  smaller
43              than 5.5*10**-10 -- that is, the resulting value of Gamma when
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45                exp( ln_Gamma( x) )
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47              is  computed  is expected to be precise to better than nine sig‐
48              nificant figures.
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50       ::math::factorial x
51              Returns the factorial of the argument x.
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53              For integer x, 0  <=  x  <=  12,  an  exact  integer  result  is
54              returned.
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56              For  integer x, 13 <= x <= 21, an exact floating-point result is
57              returned on machines with IEEE floating point.
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59              For integer x, 22 <= x <= 170, the result is exact to 1 ULP.
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61              For real x, x >= 0, the  result  is  approximated  by  computing
62              Gamma(x+1)  using  the ::math::ln_Gamma function, and the result
63              is expected to be precise to better than nine  significant  fig‐
64              ures.
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66              It  is  an  error to present x <= -1 or x > 170, or a value of x
67              that is not numeric.
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69       ::math::choose n k
70              Returns the binomial coefficient C(n, k)
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72                 C(n,k) = n! / k! (n-k)!
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74              If both parameters are integers and the result fits in 32  bits,
75              the result is rounded to an integer.
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77              Integer results are exact up to at least n = 34.  Floating point
78              results are precise to better than nine significant figures.
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80       ::math::Beta z w
81              Returns the Beta function of the parameters z and w.
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83                 Beta(z,w) = Beta(w,z) = Gamma(z) * Gamma(w) / Gamma(z+w)
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85              Results are returned as a floating point number precise to  bet‐
86              ter  than nine significant digits provided that w and z are both
87              at least 1.
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BUGS, IDEAS, FEEDBACK

90       This document, and the package it describes, will  undoubtedly  contain
91       bugs  and  other  problems.  Please report such in the category math of
92       the         Tcllib         SF         Trackers          [http://source‐
93       forge.net/tracker/?group_id=12883].   Please  also report any ideas for
94       enhancements you may have for either package and/or documentation.
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98math                                 1.2.3              math::combinatorics(n)
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